nPr and nCr — arrangements and selections from a set.

What's the difference between permutations and combinations?

Permutations count arrangements where order matters — 1st, 2nd, and 3rd place among 5 runners is a permutation, since swapping who's 1st and who's 2nd creates a different outcome. Combinations count selections where order doesn't matter — choosing any 3 of those 5 runners for a relay team is a combination, since the same 3 people chosen in any order count as one outcome.

What does n! (factorial) mean?

n! (read "n factorial") is the product of all positive integers up to n — 5! = 5×4×3×2×1 = 120. Both the permutation and combination formulas are built from factorials, though this tool computes them via a running product to avoid overflow on larger numbers.

Why does nCr equal nPr divided by r!?

Permutations count every ordering of a chosen group as a separate outcome, but combinations treat all r! possible orderings of the same group as a single outcome. Dividing nPr by r! removes exactly those duplicate orderings, leaving just the count of distinct groups.

What's a real-world example of each?

Permutations: arranging books in a specific order on a shelf, or assigning gold/silver/bronze medals. Combinations: choosing a committee of people from a larger group, or picking lottery numbers, where the order you select them doesn't change the outcome.

Worked example: for n=10, r=3: permutations (order matters) = 10×9×8 = 720. Combinations (order doesn't matter) = 720/3! = 720/6 = 120 — exactly 6 times fewer, since each group of 3 can be arranged in 3! = 6 different orders.